Optimal. Leaf size=313 \[ \frac {2 a \sin (c+d x) \sqrt {\sec (c+d x)} \left (3 a^2 (3 A+5 C)+35 a b B+24 A b^2\right )}{15 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^3 B+3 a^2 b (A+3 C)+9 a b^2 B+b^3 (3 A+C)\right )}{3 d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^3 (3 A+5 C)+15 a^2 b B+15 a b^2 (A-C)-5 b^3 B\right )}{5 d}-\frac {2 b^2 \sin (c+d x) (5 a B+9 A b-5 b C)}{15 d \sqrt {\sec (c+d x)}}+\frac {2 (5 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{15 d}+\frac {2 A \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{5 d} \]
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Rubi [A] time = 0.96, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {4221, 3047, 3031, 3023, 2748, 2641, 2639} \[ \frac {2 a \sin (c+d x) \sqrt {\sec (c+d x)} \left (3 a^2 (3 A+5 C)+35 a b B+24 A b^2\right )}{15 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (3 a^2 b (A+3 C)+a^3 B+9 a b^2 B+b^3 (3 A+C)\right )}{3 d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^3 (3 A+5 C)+15 a^2 b B+15 a b^2 (A-C)-5 b^3 B\right )}{5 d}-\frac {2 b^2 \sin (c+d x) (5 a B+9 A b-5 b C)}{15 d \sqrt {\sec (c+d x)}}+\frac {2 (5 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{15 d}+\frac {2 A \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{5 d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2748
Rule 3023
Rule 3031
Rule 3047
Rule 4221
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x) \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 A (a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {1}{5} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x))^2 \left (\frac {1}{2} (6 A b+5 a B)+\frac {1}{2} (3 a A+5 b B+5 a C) \cos (c+d x)-\frac {1}{2} b (3 A-5 C) \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 (6 A b+5 a B) (a+b \cos (c+d x))^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac {2 A (a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {1}{15} \left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x)) \left (\frac {1}{4} \left (24 A b^2+35 a b B+3 a^2 (3 A+5 C)\right )+\frac {1}{4} \left (5 a^2 B+15 b^2 B+6 a b (A+5 C)\right ) \cos (c+d x)-\frac {3}{4} b (9 A b+5 a B-5 b C) \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a \left (24 A b^2+35 a b B+3 a^2 (3 A+5 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 (6 A b+5 a B) (a+b \cos (c+d x))^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac {2 A (a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}-\frac {1}{15} \left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{8} \left (-24 A b^3-5 a^3 B-50 a b^2 B-15 a^2 b (A+3 C)\right )+\frac {3}{8} \left (15 a^2 b B-5 b^3 B+15 a b^2 (A-C)+a^3 (3 A+5 C)\right ) \cos (c+d x)+\frac {3}{8} b^2 (9 A b+5 a B-5 b C) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 b^2 (9 A b+5 a B-5 b C) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {2 a \left (24 A b^2+35 a b B+3 a^2 (3 A+5 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 (6 A b+5 a B) (a+b \cos (c+d x))^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac {2 A (a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}-\frac {1}{45} \left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {15}{16} \left (a^3 B+9 a b^2 B+b^3 (3 A+C)+3 a^2 b (A+3 C)\right )+\frac {9}{16} \left (15 a^2 b B-5 b^3 B+15 a b^2 (A-C)+a^3 (3 A+5 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 b^2 (9 A b+5 a B-5 b C) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {2 a \left (24 A b^2+35 a b B+3 a^2 (3 A+5 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 (6 A b+5 a B) (a+b \cos (c+d x))^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac {2 A (a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {1}{3} \left (\left (a^3 B+9 a b^2 B+b^3 (3 A+C)+3 a^2 b (A+3 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx-\frac {1}{5} \left (\left (15 a^2 b B-5 b^3 B+15 a b^2 (A-C)+a^3 (3 A+5 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=-\frac {2 \left (15 a^2 b B-5 b^3 B+15 a b^2 (A-C)+a^3 (3 A+5 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (a^3 B+9 a b^2 B+b^3 (3 A+C)+3 a^2 b (A+3 C)\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{3 d}-\frac {2 b^2 (9 A b+5 a B-5 b C) \sin (c+d x)}{15 d \sqrt {\sec (c+d x)}}+\frac {2 a \left (24 A b^2+35 a b B+3 a^2 (3 A+5 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 (6 A b+5 a B) (a+b \cos (c+d x))^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 d}+\frac {2 A (a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 2.24, size = 276, normalized size = 0.88 \[ \frac {\sqrt {\sec (c+d x)} \left (\frac {2}{5} a^3 A \tan (c+d x) \sec (c+d x)+\frac {2}{5} a \sin (c+d x) \left (3 a^2 A+5 a^2 C+15 a b B+15 A b^2\right )+\frac {2}{3} \sec (c+d x) \left (a^3 B \sin (c+d x)+3 a^2 A b \sin (c+d x)\right )+\frac {1}{3} b^3 C \sin (2 (c+d x))\right )}{d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (5 a^3 B+15 a^2 A b+45 a^2 b C+45 a b^2 B+15 A b^3+5 b^3 C\right )+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-9 a^3 A-15 a^3 C-45 a^2 b B-45 a A b^2+45 a b^2 C+15 b^3 B\right )}{\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}}{15 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b^{3} \cos \left (d x + c\right )^{5} + {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{4} + A a^{3} + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )\right )} \sec \left (d x + c\right )^{\frac {7}{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 11.13, size = 1419, normalized size = 4.53 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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